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Let X be a set, and W c P(X).

(1) X :- W

(2) /\B,A:-W A n B :- W

Prove that W is a base for W*.

(1) X :- W

(2) /\B,A:-W A n B :- W

Prove that W is a base for W*.

page 61 in gen top

Let X be a set, and W c P(X).

What does it mean that W is a topology?

What does it mean that W is a topology?

1) O :- W

2) X :- W

3) /\B,A:-W [ B n A :- W ]

4) /\McW [ \\//M :- W ]

2) X :- W

3) /\B,A:-W [ B n A :- W ]

4) /\McW [ \\//M :- W ]

Let X be a set, and W c P(X).

If W is a topology, then W* = ???

If W is a topology, then W* = ???

W* = W

HINT:

From ( /\B,A:-W AnB:-W ) we conclude that W is a base for W*.

That is the subject matter of another item.

page 63 in gen top

HINT:

From ( /\B,A:-W AnB:-W ) we conclude that W is a base for W*.

That is the subject matter of another item.

page 63 in gen top

Let X be a set. Let A,B c P(X).

Prove that A c B ==> A* c B*.

Prove that A c B ==> A* c B*.

page 63 in gen top

Let W c P(X).

Let Wd denote the collection of all finite intersections of sets from W.

Consider the set Wd u {X}.

What interesting feature does it have?

Let Wd denote the collection of all finite intersections of sets from W.

Consider the set Wd u {X}.

What interesting feature does it have?

It is a base for W*.

page 64 in gen top

page 64 in gen top

Let W c P(X).

Is it always true that W* is the smallest topology containing W?

Is it always true that W* is the smallest topology containing W?

Yes.

Recall that W* always is a topology.

Recall that if AcBcP(X), then A*cB*.

Recall that if Y is a topology, then Y*=Y.

Put it all together.

page 64,66 in gen top

Recall that W* always is a topology.

Recall that if AcBcP(X), then A*cB*.

Recall that if Y is a topology, then Y*=Y.

Put it all together.

page 64,66 in gen top

Let d be metric on |R.

Is it possible that (|R,d) is incomplete?

Is it possible that (|R,d) is incomplete?

Yes.

d(x,y) = | arctan(x) - arctan(y) |

lim(n->oo) arctan(n) = PI/2

Hence the sequence {arctan(n)} is Cauchy with respect to the Euclidean metric.

Hence the sequence {n} is Cauchy with respect to d.

d(x,y) = | arctan(x) - arctan(y) |

lim(n->oo) arctan(n) = PI/2

Hence the sequence {arctan(n)} is Cauchy with respect to the Euclidean metric.

Hence the sequence {n} is Cauchy with respect to d.

(1) X c |R is incomplete in |R with respect to the Euclidean
metric.

(2) (X,d) is a complete metric space

(3) (|R,r) is an incomplete metric space

(4) (|R,r) and (X,d) are homeomorphic

Is this setup possible?

(2) (X,d) is a complete metric space

(3) (|R,r) is an incomplete metric space

(4) (|R,r) and (X,d) are homeomorphic

Is this setup possible?

Let X = ] -PI/2 , PI/2 [.

Let d(y,x) = |tan(y) - tan(x)|.

Now (X,d) is a complete metric space.

Let r(x,y) = |arctan(x) - arctan(y)|.

Now (|R,r) is an incomplete metric space.

Let f : |R -> X, f(x) = arctan(x).

Now f is a homeomorphism.

Let d(y,x) = |tan(y) - tan(x)|.

Now (X,d) is a complete metric space.

Let r(x,y) = |arctan(x) - arctan(y)|.

Now (|R,r) is an incomplete metric space.

Let f : |R -> X, f(x) = arctan(x).

Now f is a homeomorphism.

If X is a set, what is an "outer measure" on X?

Y : P(X) -> [0,oo]

(1) Y(O) = 0

(2) A c B c X => Y(A) <= Y(B)

(3) Y is countably subadditive

(1) Y(O) = 0

(2) A c B c X => Y(A) <= Y(B)

(3) Y is countably subadditive

If X is a set, what is a "pseudometric" on X?

d : X x X -> |R

(1) /\a:-X d(a,a) = 0

(2) /\a,b:-X d(a,b) = d(b,a)

(3) /\a,b,c:-X d(a,c) <= d(a,b) + d(b,c)

REMARK: Maybe it's convenient to allow +oo as a possible value d(x,y)=oo.

(1) /\a:-X d(a,a) = 0

(2) /\a,b:-X d(a,b) = d(b,a)

(3) /\a,b,c:-X d(a,c) <= d(a,b) + d(b,c)

REMARK: Maybe it's convenient to allow +oo as a possible value d(x,y)=oo.